It’s kind of hard believe that two values with different dimensions are tied to each other closely. How can we prove it get a deeper understanding of
it. Googling “cube shadow theorem” doesn’t find anything useful else except these two videos. (Not sure why there’s no wikipedia entry for this.)

The area of the shadow on xy-plane is the sum of three plane, uv, vw, wu, projected onto xy-plane. Let’s focus on uv firstly; u x v gives us
a vector perpendicular to uv facet with the magnitude of its area, which is w in the unit cube case. Projecting uv facet onto xy-plane is
equivalent to calculate the dot product of these two normal vectors, w and z, which is equivalent to projecting w to z-axis. If we do the same
for the other two facets, and sum all three values, we get (u + v + w) . z.